Error

Derivation

1. Initial program 13.8

   \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\]

2. Initial simplification13.8

   \[\leadsto (\left(\frac{-1}{e^{\left|x\right| \cdot \left|x\right|}}\right) \cdot \left((\left(\frac{1}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} \cdot \frac{1}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right) \cdot \left((\left(\frac{1}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right) \cdot \left((\left(\frac{1}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right) \cdot \left(\frac{1.061405429}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + -1.453152027\right) + 1.421413741)_*\right) + -0.284496736)_*\right) + \left(\frac{0.254829592}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right))_*\right) + 1)_*\]
3. Using strategy rm

4. Applied add-log-exp13.8

   \[\leadsto \color{blue}{\log \left(e^{(\left(\frac{-1}{e^{\left|x\right| \cdot \left|x\right|}}\right) \cdot \left((\left(\frac{1}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} \cdot \frac{1}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right) \cdot \left((\left(\frac{1}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right) \cdot \left((\left(\frac{1}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right) \cdot \left(\frac{1.061405429}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + -1.453152027\right) + 1.421413741)_*\right) + -0.284496736)_*\right) + \left(\frac{0.254829592}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right))_*\right) + 1)_*}\right)}\]
5. Using strategy rm

6. Applied add-exp-log13.8

   \[\leadsto \color{blue}{e^{\log \left(\log \left(e^{(\left(\frac{-1}{e^{\left|x\right| \cdot \left|x\right|}}\right) \cdot \left((\left(\frac{1}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} \cdot \frac{1}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right) \cdot \left((\left(\frac{1}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right) \cdot \left((\left(\frac{1}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right) \cdot \left(\frac{1.061405429}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + -1.453152027\right) + 1.421413741)_*\right) + -0.284496736)_*\right) + \left(\frac{0.254829592}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right))_*\right) + 1)_*}\right)\right)}}\]

7. Final simplification13.8

   \[\leadsto e^{\log \left(\log \left(e^{(\left(\frac{-1}{e^{\left|x\right| \cdot \left|x\right|}}\right) \cdot \left((\left(\frac{1}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} \cdot \frac{1}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right) \cdot \left((\left(\frac{1}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right) \cdot \left((\left(\frac{1}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right) \cdot \left(\frac{1.061405429}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*} + -1.453152027\right) + 1.421413741)_*\right) + -0.284496736)_*\right) + \left(\frac{0.254829592}{(0.3275911 \cdot \left(\left|x\right|\right) + 1)_*}\right))_*\right) + 1)_*}\right)\right)}\]

Runtime

Time bar (total: 1.1m)Debug logProfile

herbie shell --seed 2018258 +o rules:numerics
(FPCore (x)
  :name "Jmat.Real.erf"
  (- 1 (* (* (/ 1 (+ 1 (* 0.3275911 (fabs x)))) (+ 0.254829592 (* (/ 1 (+ 1 (* 0.3275911 (fabs x)))) (+ -0.284496736 (* (/ 1 (+ 1 (* 0.3275911 (fabs x)))) (+ 1.421413741 (* (/ 1 (+ 1 (* 0.3275911 (fabs x)))) (+ -1.453152027 (* (/ 1 (+ 1 (* 0.3275911 (fabs x)))) 1.061405429))))))))) (exp (- (* (fabs x) (fabs x)))))))